Contents
P3.1 Simple pendulum
clear variables
syms th om Om g l T real
syms s
eq1=s*th-om;
eq2=s*om+g/l*th-T;
[A,B]=equationsToMatrix([eq1,eq2],[th,om]);
disp('State transition matrix:')
A=subs(A,g/l,Om^2);
eAt=ilaplace(inv(A))
disp('Response modes:')
[Om;1].*eAt(:,2)
State transition matrix:
eAt =
[ cos(Om*t), sin(Om*t)/Om]
[ -Om*sin(Om*t), cos(Om*t)]
Response modes:
ans =
sin(Om*t)
cos(Om*t)
P3.2 DC motor
syms V ia th om R L J b kt kb tau real
syms s
eq1=R*ia+kb*om-V;
eq2=(J*s+b)*om-kt*ia;
eq2=subs(eq2,ia,solve(eq1,ia));
eq3=s*th-om;
[A,B]=equationsToMatrix([eq2,eq3],[om,th]);
A=subs(A,b+kt*kb/R,J/tau);
E=sym(eye(2));E(1,1)=1/J;
A=E*A; B=E*B;
disp('State transition matrix:')
eAt=ilaplace(inv(A))
disp('Response modes:')
ilaplace(1./diag(A))
State transition matrix:
eAt =
[ exp(-t/tau), 0]
[ tau - tau*exp(-t/tau), 1]
Response modes:
ans =
exp(-t/tau)
1
P3.3 Dynamic system
syms t real
G=zpk([-1],[0 -2 -5],1);
disp('Controller form:')
canon(G,'companion')
disp('Modal form:')
G=canon(G,'modal')
disp('Response modes:')
exp(diag(G.A)*t)
Controller form:
ans =
a =
x1 x2 x3
x1 0 0 0
x2 1 0 -10
x3 0 1 -7
b =
u1
x1 1
x2 0
x3 0
c =
x1 x2 x3
y1 0 1 -6
d =
u1
y1 0
Continuous-time state-space model.
Modal form:
G =
a =
x1 x2 x3
x1 0 0 0
x2 0 -2 0
x3 0 0 -5
b =
u1
x1 0.8
x2 1.886
x3 1.521
c =
x1 x2 x3
y1 0.125 0.08839 -0.1754
d =
u1
y1 0
Continuous-time state-space model.
Response modes:
ans =
1
exp(-2*t)
exp(-5*t)
P3.4 Dynamic system
G=tf([1 1],[1 2 2 0]);
disp('Modal form:')
G=canon(G,'modal')
eAt=ilaplace(inv(s*eye(3)-G.a));
disp('Response modes:')
[eAt(1,1);eAt(2:3,3)]
Modal form:
G =
a =
x1 x2 x3
x1 0 0 0
x2 0 -1 1
x3 0 -1 -1
b =
u1
x1 0.5
x2 1.203
x3 0.7438
c =
x1 x2 x3
y1 1 -0.1148 -0.4864
d =
u1
y1 0
Continuous-time state-space model.
Response modes:
ans =
1
exp(-t)*sin(t)
exp(-t)*cos(t)
P3.5 Automobile
G=tf([28 120],[1 7 14]);
disp('Controller form:')
canon(G,'companion')
disp('Modal form:')
G=canon(G,'modal')
eAt=ilaplace(inv(s*eye(2)-G.a));
disp('Response modes:')
vpa(eAt(:,2),2)
Controller form:
ans =
a =
x1 x2
x1 0 -14
x2 1 -7
b =
u1
x1 1
x2 0
c =
x1 x2
y1 28 -76
d =
u1
y1 0
Continuous-time state-space model.
Modal form:
G =
a =
x1 x2
x1 -3.5 1.323
x2 -1.323 -3.5
b =
u1
x1 5.855
x2 0.9935
c =
x1 x2
y1 5.117 -1.972
d =
u1
y1 0
Continuous-time state-space model.
Response modes:
ans =
exp(-3.5*t)*sin(1.3*t)
exp(-3.5*t)*cos(1.3*t)
P3.6 Human postural dynamics
syms th om Om real
syms s
eq1=s*th-om;
eq2=s*om+Om^2*th-T;
[A,B]=equationsToMatrix([eq1,eq2],[th,om]);
disp('State transition matrix:')
A=subs(A,g/l,Om^2);
eAt=ilaplace(inv(A))
disp('Response modes:')
[Om;1].*eAt(:,2)
State transition matrix:
eAt =
[ cos(Om*t), sin(Om*t)/Om]
[ -Om*sin(Om*t), cos(Om*t)]
Response modes:
ans =
sin(Om*t)
cos(Om*t)